For a sales​ promotion, the manufacturer places winning symbols under the caps of 23% of all its soda bottles. If you buy a​ six-pack of​ soda, what is the probability that you win​ something?

Answer:

The probability that you win​ something is 0.7916.

Step-by-step explanation:

Let X denote the number of soda bottles having winning symbols under the caps.

The proportion of soda bottles having winning symbols under the caps is, p = 0.23.

Six-pack of soda are randomly bought.

Every bottle is independent of the others to have the symbols under the caps.

The random variable X follows a binomial distribution with parameters n = 6 and p = 0.23.

Then the chance of winning something implies that at least 1 of the bottles in the six-pack have the symbol under the caps.

Compute the value of P (X ≥ 1) as follows:

[tex]P(X\geq 1)=1-P(X<1)\\\\=1-P(X=0)\\\\=1-{6\choose 0}(0.23)^{0}(1-0.23)^{6-0}\\\\=1-0.2084224\\\\=0.7915776\\\\\approx 0.7916[/tex]

Thus, the probability that you win​ something is 0.7916.

The probability of winning something from buying 6 bottles of the considered soda company is 0.7916 approximately.

Binomial distributions consists of n independent Bernoulli trials.

Bernoulli trials are those trials which end up randomly either on success (with probability p) or on failures( with probability 1- p = q (say))

Suppose we have random variable X pertaining binomial distribution with parameters n and p, then it is written as

[tex]X \sim B(n,p)[/tex]

The probability that out of n trials, there'd be x successes is given by

[tex]P(X =x) = \: ^nC_xp^x(1-p)^{n-x}[/tex]

For this case, let we take:
X = the number of bottles out of 6 bottles having winning symbols.

Each bottle have the winning symbol(call it success) or not (call it failure).

Each bottle's result is independent of each other in a random picking (if 23% of the number of bottles manufactured is bigger or equal 6).

Then, we have X pertaining binomial distribution, as:
[tex]X \sim B(n=6, p = 0.23)[/tex]

(since success is a bottle having winning stickers, and as there are only 23% such bottles, so chance of success is 23% = 0.23(in decimal)).

Probability of not winning anything = P(X = 0)

From the probability function of binomial distribution, we get:

[tex]P(X = 0) = \: ^nC_xp^x(1-p)^{n-x} = \: ^6C_0(0.23)^0(1-0.23)^6 \approx 0.2084[/tex]

Thus, we get:

Probability of winning something = 1 - probability of not winning anything.

Probability of winning something ≈ 1- 0.2084 = 0.7916

Thus, the probability of winning something from buying 6 bottles of the considered soda company is 0.7916 approximately.

Learn more about binomial distribution here:

https://brainly.com/question/13609688

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